# Properties

 Label 245.2.e.c Level $245$ Weight $2$ Character orbit 245.e Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,2,Mod(116,245)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.116");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} - 6 q^{6} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (3*z - 3) * q^3 + (2*z - 2) * q^4 + z * q^5 - 6 * q^6 - 6*z * q^9 $$q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} - 6 q^{6} - 6 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + (\zeta_{6} - 1) q^{11} - 6 \zeta_{6} q^{12} + 3 q^{13} - 3 q^{15} + 4 \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + ( - 12 \zeta_{6} + 12) q^{18} - 6 \zeta_{6} q^{19} - 2 q^{20} - 2 q^{22} + 4 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 6 \zeta_{6} q^{26} + 9 q^{27} - q^{29} - 6 \zeta_{6} q^{30} + (6 \zeta_{6} - 6) q^{31} + (8 \zeta_{6} - 8) q^{32} - 3 \zeta_{6} q^{33} + 6 q^{34} + 12 q^{36} + ( - 12 \zeta_{6} + 12) q^{38} + (9 \zeta_{6} - 9) q^{39} + 6 q^{41} - 6 q^{43} - 2 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{45} + (8 \zeta_{6} - 8) q^{46} + 9 \zeta_{6} q^{47} - 12 q^{48} - 2 q^{50} + 9 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{52} + ( - 10 \zeta_{6} + 10) q^{53} + 18 \zeta_{6} q^{54} - q^{55} + 18 q^{57} - 2 \zeta_{6} q^{58} + ( - 6 \zeta_{6} + 6) q^{59} + ( - 6 \zeta_{6} + 6) q^{60} - 12 q^{62} - 8 q^{64} + 3 \zeta_{6} q^{65} + ( - 6 \zeta_{6} + 6) q^{66} + ( - 14 \zeta_{6} + 14) q^{67} + 6 \zeta_{6} q^{68} - 12 q^{69} - 8 q^{71} + (6 \zeta_{6} - 6) q^{73} - 3 \zeta_{6} q^{75} + 12 q^{76} - 18 q^{78} + \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} + (9 \zeta_{6} - 9) q^{81} + 12 \zeta_{6} q^{82} + 12 q^{83} + 3 q^{85} - 12 \zeta_{6} q^{86} + ( - 3 \zeta_{6} + 3) q^{87} - 12 \zeta_{6} q^{89} + 12 q^{90} - 8 q^{92} - 18 \zeta_{6} q^{93} + (18 \zeta_{6} - 18) q^{94} + ( - 6 \zeta_{6} + 6) q^{95} - 24 \zeta_{6} q^{96} - 15 q^{97} + 6 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (3*z - 3) * q^3 + (2*z - 2) * q^4 + z * q^5 - 6 * q^6 - 6*z * q^9 + (2*z - 2) * q^10 + (z - 1) * q^11 - 6*z * q^12 + 3 * q^13 - 3 * q^15 + 4*z * q^16 + (-3*z + 3) * q^17 + (-12*z + 12) * q^18 - 6*z * q^19 - 2 * q^20 - 2 * q^22 + 4*z * q^23 + (z - 1) * q^25 + 6*z * q^26 + 9 * q^27 - q^29 - 6*z * q^30 + (6*z - 6) * q^31 + (8*z - 8) * q^32 - 3*z * q^33 + 6 * q^34 + 12 * q^36 + (-12*z + 12) * q^38 + (9*z - 9) * q^39 + 6 * q^41 - 6 * q^43 - 2*z * q^44 + (-6*z + 6) * q^45 + (8*z - 8) * q^46 + 9*z * q^47 - 12 * q^48 - 2 * q^50 + 9*z * q^51 + (6*z - 6) * q^52 + (-10*z + 10) * q^53 + 18*z * q^54 - q^55 + 18 * q^57 - 2*z * q^58 + (-6*z + 6) * q^59 + (-6*z + 6) * q^60 - 12 * q^62 - 8 * q^64 + 3*z * q^65 + (-6*z + 6) * q^66 + (-14*z + 14) * q^67 + 6*z * q^68 - 12 * q^69 - 8 * q^71 + (6*z - 6) * q^73 - 3*z * q^75 + 12 * q^76 - 18 * q^78 + z * q^79 + (4*z - 4) * q^80 + (9*z - 9) * q^81 + 12*z * q^82 + 12 * q^83 + 3 * q^85 - 12*z * q^86 + (-3*z + 3) * q^87 - 12*z * q^89 + 12 * q^90 - 8 * q^92 - 18*z * q^93 + (18*z - 18) * q^94 + (-6*z + 6) * q^95 - 24*z * q^96 - 15 * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{5} - 12 q^{6} - 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 2 * q^4 + q^5 - 12 * q^6 - 6 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{5} - 12 q^{6} - 6 q^{9} - 2 q^{10} - q^{11} - 6 q^{12} + 6 q^{13} - 6 q^{15} + 4 q^{16} + 3 q^{17} + 12 q^{18} - 6 q^{19} - 4 q^{20} - 4 q^{22} + 4 q^{23} - q^{25} + 6 q^{26} + 18 q^{27} - 2 q^{29} - 6 q^{30} - 6 q^{31} - 8 q^{32} - 3 q^{33} + 12 q^{34} + 24 q^{36} + 12 q^{38} - 9 q^{39} + 12 q^{41} - 12 q^{43} - 2 q^{44} + 6 q^{45} - 8 q^{46} + 9 q^{47} - 24 q^{48} - 4 q^{50} + 9 q^{51} - 6 q^{52} + 10 q^{53} + 18 q^{54} - 2 q^{55} + 36 q^{57} - 2 q^{58} + 6 q^{59} + 6 q^{60} - 24 q^{62} - 16 q^{64} + 3 q^{65} + 6 q^{66} + 14 q^{67} + 6 q^{68} - 24 q^{69} - 16 q^{71} - 6 q^{73} - 3 q^{75} + 24 q^{76} - 36 q^{78} + q^{79} - 4 q^{80} - 9 q^{81} + 12 q^{82} + 24 q^{83} + 6 q^{85} - 12 q^{86} + 3 q^{87} - 12 q^{89} + 24 q^{90} - 16 q^{92} - 18 q^{93} - 18 q^{94} + 6 q^{95} - 24 q^{96} - 30 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 2 * q^4 + q^5 - 12 * q^6 - 6 * q^9 - 2 * q^10 - q^11 - 6 * q^12 + 6 * q^13 - 6 * q^15 + 4 * q^16 + 3 * q^17 + 12 * q^18 - 6 * q^19 - 4 * q^20 - 4 * q^22 + 4 * q^23 - q^25 + 6 * q^26 + 18 * q^27 - 2 * q^29 - 6 * q^30 - 6 * q^31 - 8 * q^32 - 3 * q^33 + 12 * q^34 + 24 * q^36 + 12 * q^38 - 9 * q^39 + 12 * q^41 - 12 * q^43 - 2 * q^44 + 6 * q^45 - 8 * q^46 + 9 * q^47 - 24 * q^48 - 4 * q^50 + 9 * q^51 - 6 * q^52 + 10 * q^53 + 18 * q^54 - 2 * q^55 + 36 * q^57 - 2 * q^58 + 6 * q^59 + 6 * q^60 - 24 * q^62 - 16 * q^64 + 3 * q^65 + 6 * q^66 + 14 * q^67 + 6 * q^68 - 24 * q^69 - 16 * q^71 - 6 * q^73 - 3 * q^75 + 24 * q^76 - 36 * q^78 + q^79 - 4 * q^80 - 9 * q^81 + 12 * q^82 + 24 * q^83 + 6 * q^85 - 12 * q^86 + 3 * q^87 - 12 * q^89 + 24 * q^90 - 16 * q^92 - 18 * q^93 - 18 * q^94 + 6 * q^95 - 24 * q^96 - 30 * q^97 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i −1.50000 + 2.59808i −1.00000 + 1.73205i 0.500000 + 0.866025i −6.00000 0 0 −3.00000 5.19615i −1.00000 + 1.73205i
226.1 1.00000 1.73205i −1.50000 2.59808i −1.00000 1.73205i 0.500000 0.866025i −6.00000 0 0 −3.00000 + 5.19615i −1.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.e.c 2
7.b odd 2 1 245.2.e.d 2
7.c even 3 1 245.2.a.b yes 1
7.c even 3 1 inner 245.2.e.c 2
7.d odd 6 1 245.2.a.a 1
7.d odd 6 1 245.2.e.d 2
21.g even 6 1 2205.2.a.j 1
21.h odd 6 1 2205.2.a.l 1
28.f even 6 1 3920.2.a.bj 1
28.g odd 6 1 3920.2.a.a 1
35.i odd 6 1 1225.2.a.j 1
35.j even 6 1 1225.2.a.h 1
35.k even 12 2 1225.2.b.b 2
35.l odd 12 2 1225.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 7.d odd 6 1
245.2.a.b yes 1 7.c even 3 1
245.2.e.c 2 1.a even 1 1 trivial
245.2.e.c 2 7.c even 3 1 inner
245.2.e.d 2 7.b odd 2 1
245.2.e.d 2 7.d odd 6 1
1225.2.a.h 1 35.j even 6 1
1225.2.a.j 1 35.i odd 6 1
1225.2.b.a 2 35.l odd 12 2
1225.2.b.b 2 35.k even 12 2
2205.2.a.j 1 21.g even 6 1
2205.2.a.l 1 21.h odd 6 1
3920.2.a.a 1 28.g odd 6 1
3920.2.a.bj 1 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + 6T + 36$$
$37$ $$T^{2}$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 9T + 81$$
$53$ $$T^{2} - 10T + 100$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 14T + 196$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 6T + 36$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$(T + 15)^{2}$$